The scaling theorem (or similarity theorem) says that if you horizontally ``stretch'' a signal by the factor in the time domain, you ``squeeze'' and amplify its Fourier transform by the same factor in the frequency domain. This is an important general Fourier duality relationship:
Theorem: For all continuous-time functions
possessing a Fourier
transform,
where
and is any nonzero real number (the abscissa stretch factor). A more commonly used notation is the following:
(3.41) |
Proof: See §B.4.
The scaling theorem is fundamentally restricted to the continuous-time, continuous-frequency (Fourier transform) case. The closest we come to the scaling theorem among the DTFT theorems (§2.3) is the stretch (repeat) theorem (page ). For this and other continuous-time Fourier theorems, see Appendix B.